3.2499 \(\int (d+e x)^3 \sqrt [4]{a+b x+c x^2} \, dx\)

Optimal. Leaf size=374 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{17/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right )}{168 c^4}+\frac{e \left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e+243 b d)+117 b^2 e^2+130 c e x (2 c d-b e)+616 c^2 d^2\right )}{630 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c} \]

[Out]

((2*c*d - b*e)*(28*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/4))/(168*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(5/4))/(9*c
) + (e*(616*c^2*d^2 + 117*b^2*e^2 - 2*c*e*(243*b*d + 56*a*e) + 130*c*e*(2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(5/4))/(630*c^3) - ((b^2 - 4*a*c)^(5/4)*(2*c*d - b*e)*
(28*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a
*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*
(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(336*Sqrt[2]*c^(17/4)*(b +
2*c*x))

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Rubi [A]  time = 1.03515, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{17/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right )}{168 c^4}+\frac{e \left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e+243 b d)+117 b^2 e^2+130 c e x (2 c d-b e)+616 c^2 d^2\right )}{630 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)^3*(a + b*x + c*x^2)^(1/4),x]

[Out]

((2*c*d - b*e)*(28*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/4))/(168*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(5/4))/(9*c
) + (e*(616*c^2*d^2 + 117*b^2*e^2 - 2*c*e*(243*b*d + 56*a*e) + 130*c*e*(2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(5/4))/(630*c^3) - ((b^2 - 4*a*c)^(5/4)*(2*c*d - b*e)*
(28*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a
*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*
(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(336*Sqrt[2]*c^(17/4)*(b +
2*c*x))

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Rubi in Sympy [A]  time = 93.7541, size = 437, normalized size = 1.17 \[ \frac{2 e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{9 c} + \frac{8 e \left (a + b x + c x^{2}\right )^{\frac{5}{4}} \left (- 7 a c e^{2} + \frac{117 b^{2} e^{2}}{16} - \frac{243 b c d e}{8} + \frac{77 c^{2} d^{2}}{2} - \frac{65 c e x \left (b e - 2 c d\right )}{8}\right )}{315 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}} \left (- 24 a c e^{2} + 13 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right )}{168 c^{4}} + \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 24 a c e^{2} + 13 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{672 c^{\frac{17}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+b*x+a)**(1/4),x)

[Out]

2*e*(d + e*x)**2*(a + b*x + c*x**2)**(5/4)/(9*c) + 8*e*(a + b*x + c*x**2)**(5/4)
*(-7*a*c*e**2 + 117*b**2*e**2/16 - 243*b*c*d*e/8 + 77*c**2*d**2/2 - 65*c*e*x*(b*
e - 2*c*d)/8)/(315*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(1/4)*(
-24*a*c*e**2 + 13*b**2*e**2 - 28*b*c*d*e + 28*c**2*d**2)/(168*c**4) + sqrt(2)*sq
rt(-(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt
(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)**2))*(-4*a*c + b**2)**(5/4)*(b*e - 2
*c*d)*(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-24*a*c*e**2 +
13*b**2*e**2 - 28*b*c*d*e + 28*c**2*d**2)*sqrt((b + 2*c*x)**2)*elliptic_f(2*ata
n(sqrt(2)*c**(1/4)*(a + b*x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(672*
c**(17/4)*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 1.62863, size = 376, normalized size = 1.01 \[ \frac{\frac{4 (a+x (b+c x)) \left (16 c^2 \left (-28 a^2 e^3+a c e \left (189 d^2+45 d e x+7 e^2 x^2\right )+c^2 x \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )-4 b^2 c e \left (c \left (315 d^2+81 d e x+13 e^2 x^2\right )-207 a e^2\right )+8 b c^2 \left (c \left (105 d^3+63 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )-a e^2 (333 d+31 e x)\right )-195 b^4 e^3+6 b^3 c e^2 (135 d+13 e x)\right )}{15 c^4}+\frac{\sqrt [4]{2} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} (b e-2 c d) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^5}}{672 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(1/4),x]

[Out]

((4*(a + x*(b + c*x))*(-195*b^4*e^3 + 6*b^3*c*e^2*(135*d + 13*e*x) - 4*b^2*c*e*(
-207*a*e^2 + c*(315*d^2 + 81*d*e*x + 13*e^2*x^2)) + 8*b*c^2*(-(a*e^2*(333*d + 31
*e*x)) + c*(105*d^3 + 63*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3)) + 16*c^2*(-28*a^2*
e^3 + a*c*e*(189*d^2 + 45*d*e*x + 7*e^2*x^2) + c^2*x*(105*d^3 + 189*d^2*e*x + 13
5*d*e^2*x^2 + 35*e^3*x^3))))/(15*c^4) + (2^(1/4)*(b^2 - 4*a*c)*(-2*c*d + b*e)*(2
8*c^2*d^2 + 13*b^2*e^2 - 4*c*e*(7*b*d + 6*a*e))*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*
((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(3/4)*Hypergeometric2F1[1/4,
3/4, 5/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c^5)/(672*(a
+ x*(b + c*x))^(3/4))

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Maple [F]  time = 0.15, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{3}\sqrt [4]{c{x}^{2}+bx+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+b*x+a)^(1/4),x)

[Out]

int((e*x+d)^3*(c*x^2+b*x+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(c*x^2 + b*x + a)^(1/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \sqrt [4]{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)**(1/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3, x)